Open problems in infinite dimensional geometry and topology

2010 ◽  
Vol 104 (2) ◽  
pp. 199-200
Keyword(s):  
Open Problems ◽  
Infinite Dimensional ◽  
And Topology

Operators with Compact Self-Commutator

1974 ◽  
Vol 26 (1) ◽  
pp. 115-120 ◽  
Author(s):  
Carl Pearcy ◽  
Norberto Salinas
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Open Problems ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Calkin Algebra ◽  
The Ideal

Let be a fixed separable, infinite dimensional complex Hilbert space, and let () denote the algebra of all (bounded, linear) operators on . The ideal of all compact operators on will be denoted by and the canonical quotient map from () onto the Calkin algebra ()/ will be denoted by π.Some open problems in the theory of extensions of C*-algebras (cf. [1]) have recently motivated an increasing interest in the class of all operators in () whose self-commuta tor is compact.

scholarly journals Large Irredundant Sets in Operator Algebras

2019 ◽  
Vol 72 (4) ◽  
pp. 988-1023
Author(s):  
Clayton Suguio Hida ◽  
Piotr Koszmider
Keyword(s):  
Von Neumann Algebras ◽  
Operator Algebras ◽  
Continuum Hypothesis ◽  
The Other ◽  
Open Problems ◽  
Von Neumann ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
The Continuum

AbstractA subset ${\mathcal{X}}$ of a C*-algebra ${\mathcal{A}}$ is called irredundant if no $A\in {\mathcal{X}}$ belongs to the C*-subalgebra of ${\mathcal{A}}$ generated by ${\mathcal{X}}\setminus \{A\}$. Separable C*-algebras cannot have uncountable irredundant sets and all members of many classes of nonseparable C*-algebras, e.g., infinite dimensional von Neumann algebras have irredundant sets of cardinality continuum.There exists a considerable literature showing that the question whether every AF commutative nonseparable C*-algebra has an uncountable irredundant set is sensitive to additional set-theoretic axioms, and we investigate here the noncommutative case.Assuming $\diamondsuit$ (an additional axiom stronger than the continuum hypothesis), we prove that there is an AF C*-subalgebra of ${\mathcal{B}}(\ell _{2})$ of density $2^{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}_{1}$ with no nonseparable commutative C*-subalgebra and with no uncountable irredundant set. On the other hand we also prove that it is consistent that every discrete collection of operators in ${\mathcal{B}}(\ell _{2})$ of cardinality continuum contains an irredundant subcollection of cardinality continuum.Other partial results and more open problems are presented.

scholarly journals GK–DIMENSION OF ALGEBRAS WITH MANY GENERIC RELATIONS

2009 ◽  
Vol 51 (2) ◽  
pp. 253-256 ◽  
Author(s):  
AGATA SMOKTUNOWICZ
Keyword(s):  
Open Problems ◽  
Infinite Dimensional ◽  
Gk Dimension ◽  
Kirillov Dimension

AbstractWe prove some results on algebras, satisfying many generic relations. As an application we show that there are Golod–Shafarevich algebras which cannot be homomorphically mapped onto infinite dimensional algebras with finite Gelfand–Kirillov dimension. This answers a question of Zelmanov (Some open problems in the theory of infinite dimensional algebras, J. Korean Math. Soc. 44(5) 2007, 1185–1195).

scholarly journals Enveloping semigroups and quasi-discrete spectrum

2015 ◽  
Vol 36 (8) ◽  
pp. 2627-2660 ◽  
Author(s):  
JUHO RAUTIO
Keyword(s):  
Discrete Spectrum ◽  
American Mathematical Society ◽  
Natural Generalization ◽  
Minimal System ◽  
Topological Groups ◽  
Open Problems ◽  
Universal System ◽  
Infinite Dimensional ◽  
Enveloping Semigroups ◽  
The One

The structures of the enveloping semigroups of certain elementary finite- and infinite-dimensional distal dynamical systems are given, answering open problems posed in 1982 by Namioka [Ellis groups and compact right topological groups. Conference in Modern Analysis and Probability (New Haven, CT, 1982) (Contemporary Mathematics, 26). American Mathematical Society, Providence, RI, 1984, 295–300]. The universal minimal system with (topological) quasi-discrete spectrum is obtained from the infinite-dimensional case. It is proved that, on the one hand, a minimal system is a factor of this universal system if and only if its enveloping semigroup has quasi-discrete spectrum and that, on the other hand, such a factor need not have quasi-discrete spectrum in itself. This leads to a natural generalization of the property of having quasi-discrete spectrum, which is named the ${\mathcal{W}}$-property.

Orthogonal Random Vectors and the Hurwitz-Radon-Eckmann Theorem

1994 ◽  
Vol 1 (1) ◽  
pp. 99-113
Author(s):  
N. Vakhania
Keyword(s):  
Hilbert Space ◽  
Nauk Sssr ◽  
Adjoint Operator ◽  
Covariance Operator ◽  
Random Vectors ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
And Topology ◽  
Dimensional Hilbert Space ◽  
Akad Nauk Sssr

Abstract In several different aspects of algebra and topology the following problem is of interest: find the maximal number of unitary antisymmetric operators Ui in with the property UiUj = –UjUi (i ≠ j). The solution of this problem is given by the Hurwitz-Radon-Eckmann formula. We generalize this formula in two directions: all the operators Ui must commute with a given arbitrary self-adjoint operator and H can be infinite-dimensional. Our second main result deals with the conditions for almost sure orthogonality of two random vectors taking values in a finite or infinite-dimensional Hilbert space H. Finally, both results are used to get the formula for the maximal number of pairwise almost surely orthogonal random vectors in H with the same covariance operator and each pair having a linear support in H ⊕ H. The paper is based on the results obtained jointly with N.P.Kandelaki (see [Vakhania and Kandelaki, Dokl. Akad. Nauk SSSR 294: 528-531, 1987, Dokl. Akad. Nauk SSSR 296: 265-266, 1988, Trudy Inst. Vychisl. Mat. Akad. Nauk Gruz. SSR 28: 11-37, 1988]).

Infinite-Dimensional Polyhedrality

2004 ◽  
Vol 56 (3) ◽  
pp. 472-494 ◽  
Author(s):  
Vladimir P. Fonf ◽  
Libor Veselý
Keyword(s):  
Banach Space ◽  
Convex Subset ◽  
General Definition ◽  
Infinite Dimensions ◽  
Open Problems ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Isomorphic Classification

AbstractThis paper deals with generalizations of the notion of a polytope to infinite dimensions. The most general definition is the following: a bounded closed convex subset of a Banach space is called a polytope if each of its finite-dimensional affine sections is a (standard) polytope.We study the relationships between eight known definitions of infinite-dimensional polyhedrality. We provide a complete isometric classification of them, which gives solutions to several open problems. An almost complete isomorphic classification is given as well (only one implication remains open).

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